Frege, hilbert, and the conceptual structure of model theory

Abstract

This paper attempts to confine the preconceptions that prevented Frege from appreciating Hilbert’s Grundlagen der Geometrie to two: (i) Frege’s reliance on what, following Wilfrid Hodges, I call a Frege‐Peano language, and (ii) Frege’s view that the sense of an expression wholly determines its reference.I argue that these two preconceptions prevented Frege from achieving the conceptual structure of model theory, whereas Hilbert, at least in his practice, was quite close to the model‐theoretic point of view.Moreover, the issues that divided Frege and Hilbert did not revolve around whether one or the other allowed metalogical notions.Frege, e.g., succeeded in formulating the notion of logical consequence, at least to the extent that Bolzano did; the point is rather that even though Frege had certain semantic concepts, he did not articulate them model‐theoretically, whereas, in some limited sense, Hilbert did